Successive oblique impactsEdexcel A-Level Further Maths: Subtopic test

1. 1
   A smooth particle of mass $0.5$ kg moves on a smooth horizontal table. Two fixed smooth vertical walls $W_1$ and $W_2$ meet at right angles: $W_1$ is parallel to $\mathbf{i}$ and $W_2$ is parallel to $\mathbf{j}$. The particle has velocity $(8\mathbf{i}+6\mathbf{j})$ m s$^{-1}$, hits $W_1$ and then hits $W_2$. The coefficient of restitution between the particle and each wall is $0.5$.

   (a)

   Find the velocity of the particle immediately after it hits $W_1$.

   [1 mark]

   • A$(4\mathbf{i}-6\mathbf{j})$ m s$^{-1}$
   • B$(8\mathbf{i}+3\mathbf{j})$ m s$^{-1}$
   • C$(8\mathbf{i}-6\mathbf{j})$ m s$^{-1}$
   • D$(8\mathbf{i}-3\mathbf{j})$ m s$^{-1}$

   (b)

   Find the velocity of the particle immediately after it then hits $W_2$.

   [1 mark]

   • A$(-8\mathbf{i}-3\mathbf{j})$ m s$^{-1}$
   • B$(-4\mathbf{i}-3\mathbf{j})$ m s$^{-1}$
   • C$(4\mathbf{i}-3\mathbf{j})$ m s$^{-1}$
   • D$(-4\mathbf{i}+3\mathbf{j})$ m s$^{-1}$

   (c)

   Find the total kinetic energy lost by the particle in the two impacts.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A smooth sphere of mass $0.2$ kg moves on a smooth horizontal surface between two fixed parallel smooth vertical walls $W_1$ and $W_2$. It hits $W_1$ with speed $12$ m s$^{-1}$, its direction of motion making an angle of $60^\circ$ with the wall, and then crosses to hit $W_2$. The coefficient of restitution between the sphere and each wall is $\frac23$.

   (a)

   Find the angle between the path of the sphere and the wall $W_1$ immediately after the first impact.

   [1 mark]

   • A$60.0^\circ$
   • B$40.0^\circ$
   • C$49.1^\circ$
   • D$68.9^\circ$

   (b)

   Find the speed of the sphere immediately after it hits $W_2$.

   [1 mark]

   • A$7.57$ m s$^{-1}$
   • B$6.00$ m s$^{-1}$
   • C$9.17$ m s$^{-1}$
   • D$6.93$ m s$^{-1}$

   (c)

   Find the total kinetic energy lost by the sphere in the two impacts.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A smooth particle $P$ of mass $0.4$ kg moves on a smooth horizontal surface with velocity $(6\mathbf{i}+8\mathbf{j})$ m s$^{-1}$. It hits a fixed smooth vertical wall $W_1$ that is parallel to $\mathbf{j}$, and then hits a fixed smooth vertical wall $W_2$ that is parallel to $\mathbf{i}$. The coefficient of restitution between $P$ and $W_1$ is $\frac13$, and between $P$ and $W_2$ is $\frac12$.

   (a)

   Find the speed of $P$ immediately after it hits $W_1$, and the angle between its path and $W_1$.

   [3 marks]

   (b)

   Find the total kinetic energy lost by $P$ in the two impacts.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A smooth sphere of mass $0.6$ kg moves on a smooth horizontal surface with velocity $(12\mathbf{i}+9\mathbf{j})$ m s$^{-1}$. It hits a fixed smooth vertical wall $W_1$ that is parallel to $\mathbf{i}$, and then a fixed smooth vertical wall $W_2$ that is parallel to $\mathbf{j}$. The coefficient of restitution between the sphere and $W_1$ is $\frac23$, and between the sphere and $W_2$ is $\frac14$.

   (a)

   Find the velocity of the sphere after each impact, and its speed after the second impact.

   [6 marks]

   (b)

   (i) Find the total kinetic energy lost in the two impacts.
   (ii) Determine whether the final path of the sphere is parallel to its original path.

   [6 marks]

   Total for question 4: 12 marks

End of questions